
theorem Th23:
  for L being Field
  for p being non-zero Polynomial of L
  for a being Element of L
  for b being non zero Element of L st not -a/b in Roots(p)
  for E being Enumeration of Roots(<%a,b%>*'p) st E = (canFS Roots(p))^<*-a/b*>
  holds (canFS Roots(<%a,b%>*'p))" * E is Permutation of dom E
  proof
    let L be Field;
    let p be non-zero Polynomial of L;
    let a be Element of L;
    let b be non zero Element of L such that
A1: not -a/b in Roots(p);
    set q = <%a,b%>;
    set e = <*-a/b*>;
    set B = BRoots(q*'p);
    set C = canFS Roots(p);
    set D = canFS Roots(q*'p);
    let E be Enumeration of Roots(q*'p) such that
A2: E = C^e;
A3: dom E = Seg(len C + len e) by A2,FINSEQ_1:def 7;
A4: len C = card Roots(p) by FINSEQ_1:93;
A5: len e = 1 by FINSEQ_1:40;
A6: card Roots(q*'p) = 1 + card Roots(p) by A1,Th15;
A7: rng D = Roots(q*'p) by FUNCT_2:def 3;
A8: rng(D") = dom D by FUNCT_1:33;
A9: dom(D") = rng D by FUNCT_1:33;
A10: dom D = Seg len D by FINSEQ_1:def 3;
A11: len D = card Roots(q*'p) by FINSEQ_1:93;
A12: Roots(q*'p) = Roots(q) \/ Roots(p) by UPROOTS:23;
A13: rng C = Roots(p) by FUNCT_2:def 3;
A14: rng e = {-a/b} by FINSEQ_1:39;
A15: Roots(q) = {-a/b} by Th10;
A16: rng(E) = rng C \/ rng e by A2,FINSEQ_1:31;
     then
A17: rng(D"*E) = rng(D") by A9,A12,A13,A14,A15,RELAT_1:28;
    dom(D"*E) = dom(E) by A7,A9,A12,A13,A14,A15,A16,RELAT_1:27;
    then D"*E is Function of dom E,dom E
    by A3,A4,A5,A6,A8,A10,A11,A17,FUNCT_2:1;
    hence D"*E is Permutation of dom E
    by A3,A4,A5,A6,A8,A10,A11,A17,FUNCT_2:57;
  end;
